Earliest Known Uses of Some of the Words of Mathematics (R)

Last revision: June 25, 2017

RADIAN. According to Cajori (1919, page 484):

An isolated matter of interest is the origin of the term
’sradian’s, used with trigonometric functions. It first appeared in
print on June 5, 1873, in examination questions set by James Thomson
at Queen’s College, Belfast. James Thomson was a brother of Lord
Kelvin. He used the term as early as 1871, while in 1869 Thomas
Muir, then of St. Andrew’s University, hesitated between ’srad’s,
’sradial’s and ’sradian’s. In 1874, T. Muir adopted ’sradian’s after a
consultation with James Thomson.

In a letter appearing in the April 7, 1910, Nature, Thomas
Muir wrote: "I wrote to him [i.e., to Alexander J. Ellis, in 1874],
and he agreed at once for the form ’sradian,’s on the ground that it
could be viewed as a contraction for ’sradial angle’s..."

In a letter appearing in the June 16, 1910, Nature, James
Thomson wrote: "I shall be very pleased to send Dr. Muir a copy of my
father’s examination questions of June, 1873, containing the word
’sradian.’s ...It thus appears that ’sradian’s was thought of
independently by Dr. Muir and my father, and, what is really more
important than the exact form of the name, they both independently
thought of the necessity of giving a name to the unit-angle" [Dave
Cohen].

According to W. N. Roseveare, “The radian is an uninteresting angle—Lord Kelvin introduced
the word merely as a convenience in lecturing, to avoid the
long phrase ‘angle whose circular measure is.’” This quotation appears on page 133 of
W. N. Roseveare, “On ‘circular measure’ and the product forms of
the sine and cosine,” Mathematical Gazette 3 #49 (January 1905),
129-137. [Dave L. Renfro]

A post on the Internet indicated that Thomas Muir (1844-1934) claimed
to have coined the term in 1869, and that Muir and Ellis proposed the
term as a contraction of "radial angle" in 1874. A reference given
was: Michael Cooper, "Who named the radian?", Mathematical
gazette 76, no. 475 (1992) 100-101. I have not seen this article.

A 1991 Prentice-Hall high school textbook, Algebra 2, by
Bettye C. Hall and Mona Fabricant has: "James Muir, a mathematician,
and James T. Thomson, a physicist, were working independently during
the late nineteenth century to develop a new unit of angle
measurement. They met and agreed on the name radian, a
shortened form of the phrase radial angle. Different names
were used for the new unit until about 1900. Today the term
radian is in common usage."

In 1876-79, the Globe encyclopaedia of universal information has, in the Circle article:
"The unit, called a radian by Professor James Thomson, is that angle whose subtending
arc is equal in length to the radius" [University of Michigan Digital Library].

RADICAL. The word radical was used in English before
1668 by Recorde and others to refer to an irrational number.

RADICAL SIGN appears in English in 1669 in An Introduction
to Algebra edited in 1668 by John Pell (1611-1685):

In the quotient subjoyn the surd part with its first
radical Sign.

This work had earlier been translated by Thomas Branker (1636-1676),
from the original by J. H. Rahn, first published in 1659 in German.

RADICAND is found in 1889 in George Chrystal, Algebra (ed. 2) I. x. 182:
"We shall restrict the radicand, k, to be positive" (OED On Line).

Radicand also appears in an 1890 Funk and Wagnalls
dictionary.

RADIOGRAM was one of several terms launched by Karl Pearson in his lectures of 1892. According to
Stigler (1986, p. 327), “The syllabi show an abundance of fancy terminology:
stigmograms, euthygrams, epipedograms, hormograms, topograms, stereograms,
radiograms, and isodemotic lines.” Pearson soon lost interest in these
diagrams, or in all but one of them—the HISTOGRAM. Some of the
words have been re-invented, thus in Britain stereogram and radiogram became names for
types of audio equipment! [James A. Landau].

RADIUS. In modern English radius is used both for
the line joining the center of a circle to any point on its circumference and
for the length of that line. The Greeks did not have a special term for radius in either sense;
contrast DIAMETER. In Euclid Book 1, postulate 3 the term radius was not used as it is in this
translation.
The term for “distance” was used, that being thought sufficient
(Heath vol. 1 p. 199 and Smith vol. 2, page 278). Archimedes called the radius
"ek tou kentrou" (the [line] from the center) [Samuel S. Kutler].

The term “semidiameter” appears in Latin in the Ars Geometriae of Boethius (c. 510),
according to Smith (vol. 2, page 278). The English word appears in 1551 in Pathway to Knowledge by Robert Recorde:
“Defin., Diameters, whose halfe, I meane from the center to the
circumference any waie, is called the semidiameter, or halfe diameter” (OED).

The OED‘s earliest reference to radius as a mathematical term in English
is Hobbes writing in 1656, “Is the radius that
describes the inner circles equal to the radius that describes the exterior?” Six lessons to the professors of
mathematicks of the institution of Sir H. Savile, in the University of Oxford, Works. 1845 VII. 256.

[This entry was contributed by John Aldrich.]

RADIUS OF CONVERGENCE is found in English in 1891 in a translation
by George Lambert Cathcart of the German An introduction to the study of the
elements of the differential and integral calculus
by Axel Harnack [University of Michigan Historical Math Collecdtion].

Interval of convergence is found in 1891 in
An Introduction to the Study of the Elements of the Differential and Integral Calculus
by Axel Harnack. [Google print search]

RADIUS OF CURVATURE. In his Introductio in analysin
infinitorum (1748), Euler works with the radius of curvature and
says that this is commonly called "radius of osculation" but also
sometimes "radius of curvature." William C. Waterhouse provided this
citation and points out that the idea and term were in use earlier.

Thomas Simpson (1710-1761) wrote, "An equation between the radius of
curvature . . . and the angle it makes with a given direction,
implies all the conditions of the form of the curve, though not of
its position."

Radius of curvature appears in 1753 in Chambers Cyclopedia
Supplement: "Curvature, This circle is called the circle of
curvature..and its semidiameter, the ray or radius of curvature"
[OED].

The term radius of curvature may have been used earlier by
Christiaan Huygens and Isaac Newton, who wrote on the subject.

RADIX, ROOT, UNKNOWN, SQUARE ROOT. Late Latin writers used
res for the unknown. This was translated as cosa in
Italian, and the early Italian writers called algebra the Regola
de la Cosa, whence the German Die Coss and the English
cossike arte (Smith vol. 2, page 392).

Other Latin terms used in the Middle Ages for the uknown quantity and
its square were radix, res, and census.

The term root was used by al-Khowarizmi; the word is rendered
radix in Robert of Chester’s Latin translation of the algebra
of al-Khowarizmi. Radix also is used in translations from
Arabic to Latin by John of Seville, Gerard of Cremona, and Leonardo
of Pisa. For an early English use of root, see
addition.

Root (meaning "square root" or "cubic root" etc.) is found in
English in 1557 in The whetstone of witte by Robert Recorde:
"Thei onely haue rootes, whiche bee made by many multiplications of
some one number by it self" [OED].

Square root is found in English in 1557 in The whetstone of
witte by Robert Recorde: "The roote of a square nombere, is
called a Square roote" [OED].

Radix, meaning "root," appears in English in 1571 in A
geometrical practise, named Pantometria by Leonard Digges: "The
Radix Quadrate of the Product, is the Hypothenusa" [OED].

Unknown was used by Fermat. In "Novus Secundarum et
Ulterioris Ordinis Radicum in Analyticis Usus," Fermat wrote (in
translation):

There are certain problems which involve only one
unknown, and which can be called determinate, to distinguish them
from the problems of loci. There are certain others which involve
two unknowns and which can never be reduced to a single one; these
are the problems of loci. In the first problems we seek a unique
point, in the latter a curve. But if the proposed problem involves
three unknowns, one has to find, to satisfy the question, not only a
point or a curve, but an entire surface.

[Oeuvres v.1, p. 186-7, v.3, p. 161-2]

Unknown is found in English in 1676 in Glanvill, Ess.:
"The degree of Composition in the unknown Quantity of the Æquation"
[OED].

In Miscellanea Berolinensia (1710) Leibniz used the phrase
"incognita, x,."

Root (meaning "unknown") is found in English in 1728 in
Chambers Cyclopedia: "The Root of an Equation, is the Value of
the unknown Quantity in the Equation."

In the 1939 movie The Wizard of Oz, the Scarecrow
says, “The sum of the square roots of any two
sides of an isosceles triangle is equal to the square root of the
remaining side.”

RADIX (base in logarithms).Elements of Navigationby John
Robertson (1754) has: "The number usually set down, as Napier’s
logarithm of 10,
is more properly his logarithm divided by the radius of his
trigonometrical table, which is the radix from which he raised his
logarithms."
[Alan Hughes, associate editor of the OED]

RANDOM DISTRIBUTION is found in 1854 in An Investigation of the Laws of Thought
by George Boole:
"If they have not, we may regard the phaenomenon of a double star as the
accidental result of a ’srandom distribution’s of stars over the celestial vault,
i.e. of a distribution which would render it just as probable that either member of the
binary system should appear in one spot as in another." [Google print search]

Random number appears in the first book of random numbers, by Tippett:
"In order to form this table
of random numbers 40,000 digits were taken at random from census reports and
combined by fours to give 10,000 numbers." p. iii of L. H. C. Tippett’s "Random
Sampling Numbers," Tracts for Computers, No. 15 (1927).
The work was done in Karl Pearson’s Biometric Laboratory where previously random numbers
used in "experimental sampling" were generated by physically drawing marked
tickets from a bag or bowl. [OED2, James A. Landau].

Random number now usually refers to what was originally called a pseudo-random
number. The OED’s earliest quotation for pseudo-random number
is from 1949, Seminar on Sci. Computation, Nov. (Internat. Business Machines) 104/2
"A random number c lying between 0 and 1 is selected from a store, or a pseudo~random
number c lying between 0 and 1 is computed arithmetically."

See the entry MONTE CARLO.

RANDOM PROCESS. A Google search finds this phrase in 1802 in
Observations and reflections on Storms, and some other Phenomena of the Atmosphere. Letter from Professor Waterhouse to Dr. Mitchell.
The Universal Magazine of Knowledge and Pleasure, Volume 111 (1802) p 290-292.
“They seem also to have known that mountains made a part of this grand apparatus,
and to have believed that it was not a fortuitous or random process, but regulated, as we now kind it, by weight and measure.”
[Herbert Acree]

RANDOM SAMPLE.Random sample is found in 1847 in Introduction
to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries
by Henry Hallam: "This is a random sample of Feltham’s style..." [Google print search]

Random sample and random selection are found in
“A theory of probable Inference” by C. S. Pierce
which appears in the 1883 book Studies in Logic
by Members of the Johns Hopkins University and edited by Charles Sanders Peirce.

In such as case, it is our
interest that fulfils the function of an apparatus for
random selection; and no better need be desired, so
long as we have reason to deem the premise “the proportion ρ
of the M’s are P’s” to be equally true in
regard to that part of the M’s which are alone likely
ever to excite our interest. (Page 130)

In point of fact P’, P”, P’’’, etc. constitute a
random sample of the characters of M, and the ratio r
of them being found to belong to S, the same ratio of all
the characters of M are concluded to belong to S.
(Page 141)

[Google print search, Herbert Acree]

Random sampling is found in a review of “Studies in Logic” in Science,
Vol 1, No. 18 (1883), p 514-516.

When we conclude that continuous random sampling of a given natural class must leads us
towards discovering the true proportion of cases of the presence of a predestinated character in
individuals of the class; must we not base our conclusion on
the ultimate a priori assumption that our instinctive tendencies to observe natural
facts are such as, in the longrun, will lead us to actual choice at random,
and not to a choice unconsciously vitiated by unknown preference for
cases that favor the conclusion we reach?

Random sampling was used by
Karl Pearson in 1900 in the title, "On the criterion that a given system
of deviations from the probable in the case of a correlated system of variables
is such that it can be reasonably supposed to have arisen from random sampling,"
Philosophical Magazine, 50, 157-175 [OED].

Random sample is found in Karl
Pearson’s "On the Probable Errors of Frequency Constants," Biometrika,2, (1903) 273: "If the whole of a population were taken we should
have certain values for its statistical constants, but in actual practice we
are only able to take a sample, which should if possible be a random sample."
[OED].

Random variable is found in 1934 in A. Winter, “On Analytic Convolutions of Bernoulli
Distributions,” American Journal of
Mathematics, 56, 659-663 and more visibly in H. Cramér’s Random Variables and Probability Distributions (1937) (David, 1998).
Other, perhaps better, terms, including chance variable in Doob Annals of Mathematical Statistics, 6, (1935), p. 160 and
stochastic variable in Wald &
Wolfowitz, Annals of Mathematical Statistics, 10, (1939), p. 106 did not survive.
J. L. Doob recalled the time when he was writing Stochastic Processes and W. Feller was writing his Introduction to Probability Theory and its
Applications:

I had an argument with Feller.
He asserted that everyone said “random variable” and I asserted that everyone
said “chance variable.” We obviously had to use the same name in our books, so
we decided the issue by a stochastic procedure. That is, we tossed for it and
he won.

RANDOM WALK. Karl Pearson posed "The Problem of the Random Walk," in the July 27, 1905, issue of
Nature (vol. LXXII, p. 294). "A man starts from a point O and walks l yards in a straight
line; he then turns through any angle whatever and walks another l yards in a second straight line. He
repeats this process n times. I require the probability that after these stretches he is at a distance
between r and r + dr from his starting point O." Pearson’s objective was to
develop a mathematical theory of random migration. In the next issue (vol. LXXII, p. 318) Lord Rayleigh
translated the problem into one involving sound, "the composition of n iso-periodic vibrations of
unit amplitude and of phases distributed at random," and reported that he had given the solution for large
n in 1880.

The term RANDOMIZATION TEST appears in G. E. P. Box & S. L. Andersen
"Permutation Theory in the Derivation of Robust Criteria and the Study of Departures from Assumption,"
Journal of the Royal Statistical Society. Series B, 17,
(1955), p. 3. They give the term as an alternative to PERMUTATION TEST.

(David 2001)

RANDOMIZED RESPONSE. The term and the technique were introduced by Stanley L. Warner “Randomized
Response: A Survey Technique for Eliminating Evasive Answer Bias,” Journal of the American Statistical Association, 60, (1965), 63-69.

RANGE (in statistics) is found in 1848 in H. Lloyd, "On
Certain Questions Connected with the Reduction of Magnetical and
Meteorological Observations," Proceedings of the Royal Irish
Academy, 4, 180-183 (David, 1995).

RANGE (of a function).
In 1865, The Differential Calculus by John Spare has:
"It is useful to become acquainted with the methods of fully examining the entire history
of a function of one or more variables, in respect to the range of values which the function
and its variable may sustain, and to their mutual dependence" [University of Michigan Digital Library].

Range (of a function) is found in 1914 in A. R. Forsyth,
Theory of Functions of Two Complex Variables iii. 57:
"A restricted portion of a field of variation is called a domain,
the range of a domain being usually indicated by analytical relations"
[OED].

RANK (of a determinant or matrix) was coined by F. G. Frobenius, who used the German word
Rang in his paper "Uber homogene totale Differentialgleichungen,"
J.
reine angew. Math. Vol. 86 (1879) p.1.
This is according to C. C. MacDuffee, The Theory of Matrices, Springer (1933). Frobenius was defining the rank
of a determinant but the term travelled.

In English, rank (of a matrix) is found in the monograph
"Quadratic forms and their classification by means of invariant
factors", by T. J. Bromwich, Cambridge UP, 1906. This citation was
provided by Rod Gow, who writes that it is possible that an earlier
book c. 1900 by G. B. Mathews, a revision of R. F. Scott’s 1880 book
on determinants, contains the word.

Rank is also found in 1907 in Introduction
to Higher Algebra by Maxime Bôcher: where it is defined in the same
way as in Frobenius:

Definition 3. A matrix is said to be of rank r if it
contains at least one r-rowed determinant which is not zero, while
all determinants of order higher than r which the matrix may contain
are zero. A matrix is said to be of rank 0 if all its elements are
0. ... For brevity, we shall speak also of the rank of a determinant,
meaning thereby the rank of the matrix of the
determinant.

RANK CORRELATION. In his "The Proof and Measurement of Association between Two
Things," American Journal of Psychology, 15, (1904), 72-101
Charles Spearman suggested
that correlation between ranks be used as a measure of dependence between variables.
He developed the suggestion in "’sFoot-rule’s for Measuring Correlation,"
British Journal of Psychology, 2, (1906), 100-101.

Karl Pearson used the term rank correlation when he
criticised Spearman’s method in "Further Methods in Correlation," Drapers’s
Company Res. Mem. (Biometric Ser.) IV. (1907) p. 25: "No two rank correlations
are in the least reliable or comparable unless we assume that the frequency
distributions are of the same general character .. provided by the hypothesis
of normal distribution. ... Dr. Spearman has suggested that rank in a series
should be the character correlated, but he has not taken this rank correlation
as merely the stepping stone..to reach the true correlation" [OED].

[This entry was contributed by John Aldrich.]

RAO-BLACKWELL THEOREM and RAO-BLACKWELLIZATION in the theory of statistical estimation.
The "Rao-Blackwell theorem" recognises independent work by
C. R. Rao
(1945 Bull. Calcutta Math. Soc. 37, 81-91) and
David Blackwell
(1947 Ann. Math. Stat., 18, 105-110). The name dates from the 1960s for previously the theorem
had been referred to as "Blackwell’s theorem" or the "Blackwell-Rao
theorem." The term "Rao-Blackwellization" appears in Berkson
(J. Amer. Stat Assoc. 1955) ((From David (1995).)

In an ET Interview
(p. 346) Rao shares some reminiscences about getting his name attached to the result,
which may reflect more generally on the practice of EPONYMY.
When Rao objected to Berkson’s use of
Blackwellization Berkson replied that Raoization by itself "does
not sound nice." The other memory was of an exchange with D. V. Lindley who
had attributed the result to Blackwell. When Rao wrote to Lindley pointing out
his priority, Lindley replied, "Yes, I read your paper. Although the result was
in your paper, you did not realize its importance because you did not mention
it in the introduction to your paper." Rao replied, saying that it was his first
full-length paper and that he did not know that the introduction is written
for the benefit of those who read only the introduction and do not go through
the paper!

RATIO and PROPORTION. The Latin word ratio is
usually translated "computation" or "reason." St. Augustine of Hippo
(354-430) used the phrase ratio numeri in De civitate
Dei, Book 11, Chapter 30. The phrase is translated "science of
numbers" or "theory of numbers."

According to Smith (vol. 2, page 478), ratio "is a Latin word
which was commonly used in the arithmetic of the Middle Ages to mean
computation.

According to Smith (vol. 2, page 478), "To represent the idea which
we express by the symbols a:b the medieval Latin writers generally
used the word proportio, not the word ratio; while for
the idea of an equality of ratio, which we express by the symbols a:b
= c:d, they used the word proportionalitas."

In De numeris datis Jordanus (fl. 1220) wrote (in
translation), "The denomination of a ratio of this to that is what
results from dividing this by that," according to Michael S. Mahoney
in "Mathematics in the Middle Ages."

Proportion appears in 1328 in the title of the treatise De
proportionibus velocitatum in motibus by Thomas Bradwardine
(1290?-1349).

In about 1391, Chaucer wrote in English, "Abilite to lerne sciencez
touchinge noumbres & proporciouns" in Treatise on the
Astrolabe [OED].

In English, the word reason was used to mean "ratio" by
Chaucer and later by Billingsley in his 1570 translation of
Euclid’s Elements [OED].

In 1551 Robert Recorde wrote in Pathway to Knowledge:
"Lycurgus .. is most praised for that he didde chaunge the state of
their common wealthe frome the proportion Arithmeticall to a
proportion geometricall" [OED].

Ratio was used in English in 1660 by Isaac Barrow in
Euclid: Ratio (or rate) is the mutual habitude or respect of
two magnitudes of the same kind each to other, according to quantity"
[OED].

RATIONAL AND IRRATIONAL. The Greek word arrhetos dates back to at least the fourth century
B.C., and also appears in Plato. It originally meant "unspeakable,"
perhaps illustrating the Greeks’s discomfort with the idea of
irrational numbers, but it later became used to mean "irrational,"
both in Euclid’s sense and also in how we know it to be today.
The term rhetos or "rational" was later
created in contrast to arrhetos.
[Information from Dr. Gregory Dresden, consulting
"The Discovery of Incommensurability by Hippasus of Metapontum" by Kurt von Fritz and
A Greek-English Lexicon by Liddell and Scott.]

According to G. A. Miller in Historical Introduction
to Mathematical Literature (1916):

It shoud be noted that Euclid employed the terms rational [Greek spelling] and irrational
[Greek spelling] with somewhat different meanings from those now assigned to them as defined at the
beginning of this section. To explain the meaning assigned to these terms by Euclid, let a and
b be rational numbers in the modern sense, and suppose that b is not a perfect square.
According to Euclid’s definition the [sqrt b] is rational but a + [sqrt b] is irrational.
That is, while the side of a square whose area is commensurable is incommensurable in length,
Euclid says that this side is commensurable in power and considers it as rational.

Cajori (1919, page 68) writes, "It is worthy of
note that Cassiodorius was the first writer to use the terms
’srational’s and ’sirrational’s in the sense now current in arithmetic
and algebra."

Irrational is used in English by Robert Recorde in 1551 in
The Pathwaie to Knowledge: "Numbres and quantitees surde or
irrationall."

The first citation of rational in the OED2 is by John Wallis
in 1685 in Alg.: "A Fraction (in Rationals) less than the
proposed (Irrational) p."

RATIONAL FUNCTION. Euler used the term functio fracta
in his Introductio in Analysis Infinitorum (1748).

Rational function is found in English in in 1831 in the second
edition of Elements of the Differential Calculus (1836) by
John Radford Young: "The rational function may, however, become a
maximum or a minimum for more values of x than the original
root; indeed, all values of x which render the rational
function negative will render every even root of it imaginary;
such values, therefore, do not belong to that root; moreover, if the
rational function be = 0, when a maximum, the corresponding value of
the variable will be inadmissible in any even root, because the
contiguous values of the function must be negative" [James A.
Landau].

RATIO TEST. The term Cauchy’s ratio test appears in
Edward B. Van Vleck, "On Linear Criteria for the Determination of the
Radius of Convergence of a Power Series," Transactions of the
American Mathematical Society 1 (Jul., 1900).

RAYLEIGH DISTRIBUION.Lord
Rayleigh derived this distribution
as the amplitude resulting from the addition of harmonic oscillations and presented it
in "On the Resultant of a Large Number of Vibrations of the Same Pitch and of Arbitrary Phase," Philosophical
Magazine, 5th series, 10, (1880), 73-78. Rayleigh later referred to the paper
when Karl Pearson posed the problem of the random walk. See RANDOM WALK.

Whittaker and Robinson’s Calculus of Observations (second edition,
1926) has a section on the Rayleigh-Ritz
Method for Minimum Problems: “To solve this problem, Rayleigh in
1870 and subsequent years devised the following method, which was afterwards
elaborated in a celebrated memoir by W. Ritz.” The minimisation problem solved
amounts to calculating the eigenvalues associated with a differential equation.

Rayleigh quotient appears in a paper by Collatz on ODE eigenvalue
problems in 1939. Rudolf Zurmühl used it for matrices in the first
edition of his book Matrizen in 1950.

The use of the name REAL ANALYSIS to refer to a subject, a course
or book appears to have been a response to the growing use of the term COMPLEX ANALYSIS.
The popularity of the latter dates from the 1950s while
real analysis entered currency in the 1960s.

REAL NUMBER was introduced by Descartes in French in 1637.
See the entry imaginary.

The term REAL PART was used by Sir William Rowan Hamilton in
an 1843 paper. He was referring to the vector and scalar portions
of a quaternion [James A. Landau].

Real part also occurs in 1846 in W. R. Hamilton, Phil.
Mag. XXIX. 26: "The algebraically real part may receive,
according to the question in which it occurs, all values contained on
the one scale of progression of numbers from negative to positive
infinity; we shall call it therefore the scalar part, or simply the
scalar of the quaternion, and shall form its symbol by prefixing, to
the symbol of the quaternion, the characteristic Scal., or simply S"
[OED].

RECIPROCAL appears in English in 1570 in in Sir Henry
Billingsley’s translation of Euclid’s Elements: "Reciprocall
figures are those, when the termes of proportion are both antecedentes
and consequentes in either figure."

Reciprocal occurs in English, referring to quantities whose product
is 1, in the Encyclopaedia Britannica in 1797.

The term RECTANGULAR COORDINATES occurs in 1812-16 in
Playfair, Nat. Phil. (1819) II. 267: "The Sun .. and .. two
planets referred to the plane of the ecliptic, each by three
rectangular co-ordinates..parallel to the three axes" [OED].

Rectangular coordinates also appears in a paper published by
George Green in 1828 [James A. Landau].

RECTANGULAR DISTRIBUTION. See UNIFORM DISTRIBUTION.

RECTIFY (to equate a curve with a line segment). The OED shows a
use in 1673 by William Brouncker in Phil. Trans. VIII. 6150:
“It was easier to infer, That, if we can Rectifie the one, we may square the other.“

In a letter to Leibniz in 1693, Newton used rectificationem.

Rectification is found in English in 1707 in Glossographia Anglicana Nova by Thomas Blount:
“Rectification of Curves, in Mathematicks, is the assigning or finding a streight Line equal to a curved one.”
[Greg Byrnes]

RECURSION FORMULA.Recursionsformel appears in German
in 1871 in Math. Annalen IV. 113 [OED].

Recursion formula appears in English in 1905 in volume I of
The Theory of Functions of Real Variables by James Pierpont
[James A. Landau].

RECURSIVELY ENUMERABLE SET. According to Robert I. Soare
["Computability and Recursion," Bull. symbolic logic, vol. 2
(1996), p. 300] this term debuted in Alonzo Church’s "An unsolvable
problem of elementary number theory," Amer. J. Math., vol. 58
(1936), pp. 345-363. For Soare, this is "the first appearance of
’srecursively’s as an adverb meaning ’seffectively’s or ’scomputably’s."
Subsequently and in that same year the term was adopted by J. B.
Rosser in another important paper ["Extension of some theorems of
Gödel and Church," Jour. symbolic logic, vol. 1 (1936),
pp. 87-91] - and this is probably the second occurrence of the
term in the literature. It is worth mentioning also that S. C.
Kleene says in his book Mathematical Logic (1967) that "Such sets
were first considered in" his paper "General recursive functions of
natural numbers," Math. Ann., vol. 112 (1936), pp. 727-742 -
in which, in fact, the term "recursive enumeration" appears, but in
connection with functions; no term for the corresponding
sets is introduced. The inclusion of the empty set
(neglected by Kleene, Rosser and Church) was first made by Emil Post
in "Recursively enumerable sets of positive integers and their
decision problems," Bull. Amer. Math. Soc., vol 50 (1944), pp.
284-316. Recursively enumerable sets are now considered "the soul of
recursion theory" and Post’s paper was undoubtedly responsible for
this.

[This entry was contributed by Carlos César de Araújo.]

REDUCE (a fraction) is found in English in 1579 in
Stratioticos by Thomas Digges: "The Numerator of the last
Fragment to be reduced" [OED].

Abbreviate is found in 1796 in Mathem. Dict.: "To
abbreviate fractions in arithmetic and algebra, is to lessen
proportionally their terms, or the numerator and denominator" [OED].

Some writers object to the phrase "reduce a fraction" since the
fraction itself is not reduced (made smaller), although the numerator
and denominator are made smaller. They sometimes prefer the phrase
"simplify a fraction."

REDUCTIO AD ABSURDUM. This form of argument was used by Greek mathematicians and
analysed by the Greek logicians. The proof the irrationality of √2 is in
Proposition
9 of Book X of Euclid’s Elements.
The work of Euclid and Aristotle is considered in Chapter IX
§ 6. Other
Technical terms of Heath’s edition of the Elements.

Reductio ad impossibile is found in English in 1552 in T. Wilson, Rule of Reason
(ed. 2) f. 56: “The
other croked waye (called of the Logicians, Reductio ad impossibile) is a
reduccion to that, whiche is impossible” (OED).

Reductio ad absurdum is found in 1730-6 in Bailey (folio): "Exhaustions (in Mathematics) a way of
proving the equality of two magnitudes by a reductio ad absurdum; shewing that
if one be supposed either greater or less than the other, there will arise a
contradiction" (OED).

REFLEX ANGLE. An earlier term was re-entering or
re-entrant angle.

Re-entering angle appears in Phillips in 1696: "Re-entering
Angle, is that which re-enters into the body of the place" [OED].

Re-entrant angle appears in 1781 in Travels Through
Spain by Sir John T. Dillon: "He could find nothing which seemed
to confirm the opinion relating to the salient and reentrant angles"
[OED].

The 1857 Mathematical Dictionary and Cyclopedia of Mathematical
Science has re-entering angle: "RE-ENTERING ANGLE of a
polygon, is an interior angle greater than two right angles."

Reflex angle appears in 1876 in Syllabus of Plane Geometry, 2nd ed.,
by the Association for the improvement of geometrical teaching:
"A reflex angle is a term sometimes used for a major-conjgate angle."
[Google print search]

REFLEXIVE. (Of a binary relation) The OED cites Bertrand Russell writing in 1903 Principles
of Mathematics xix. p. 159 "All kinds of equality have in common the three properties of being reflexive,
symmetrical, and transitive."

REGRESSION. The statistical concept
of regression has its origins in an attempt by Francis Galton (1822-1911) to
find a mathematical law for one of the phenomena of heredity. His model (as
it would be called today) was extended by Karl Pearson and G. Udny Yule and
the biological reference eventually disappeared. The Pearson-Yule notion of
regression was based on the multivariate normal distribution but R. A. Fisher
re-founded regression using the model Gauss had proposed for the theory of errors
and method of least squares. The Pearson-Yule and Gauss-Fisher notions are still
in use. See BIOMETRY and ERROR.

The phenomenon Galton wanted to capture was reversion.
Reversion occurs when an offspring resembles an ancestor
more than its immediate parents.
The state of knowledge on the subject in Galton’s time is summarised
in chapter XIII of Charles Darwin’s
Variation
of Animals and Plants under Domestication (first edition 1869).

Galton’s model appears in the Appendix (p. 532) to his
"Typical
laws of heredity,"Nature
15 (1877), 492-495, 512-514, 532-533. Galton here focussed on the inheritance
of measurable characteristics; his observations are on the weight of peas. The
key idea is that the offspring does not inherit all the peculiarities of the
parents but is pulled back to the average of its ancestors. The idea is expressed
in what would now be called a stable first-order normal autoregressive process
where "time" is measured in generations. The process is stable because the reversion
coefficient is the fraction of the parental deviation that is inherited.

Galton continued the search for typical laws of heredity and
inspired Karl Pearson to do likewise. After the ’srediscovery’s of Mendel in 1900
much effort went into reconciling their statistical laws with his theory. Several
writers contributed but the decisive reconciliation was R.A. Fisher’s
The
Correlation Between Relatives on the Supposition
of Mendelian InheritanceTransactions of the Royal Society of
Edinburgh, 52, (1918), 399-433.

Possibly [Galton] simply felt that
["regression"] expressed more accurately the fact that offspring returned only
part way to the mean. More likely, the change reflected his new conviction,
first expressed in the same papers in which he introduced the term "regression,"
that this return to the mean reflected an inherent stability of type, and not
merely the reappearance of remote ancestral gemmules.

For Galton, regression/reversion, unlike correlation (q.v.)
another of his inventions, existed only in this biological context. The same
might have been true for Karl Pearson. However Pearson’s first major paper on
heredity ("Regression, Heredity, and Panmixia," Phil. Trans. R.
Soc., Ser. A. 187, 253-318) contains the passage:

The coefficient of regression may
be defined as the ratio of the mean deviation of the fraternity from the mean
off-spring to the deviation of the parentage from the mean parent. ... From
this special definition of regression in relation to parents and offspring,
we may pass to a general conception of regression. Let A and B be two correlated
organs (variables or measurable characteristics) in the same or different individuals,
and let the sub-group of organs B, corresponding to a sub-group of A with a
definite value a, be extracted. Let the first of these sub-groups be termed
an array, and the second a type. Then we define the coefficient of regression
of the array on the type to be the ratio of the mean-deviation of the array
from the mean B-organ to the deviation of the type a from the mean A-organ.

The biological terminology of "organs" etc. had gone
when G. U. Yule presented his boss’s theory in "On the Theory of Correlation"
(J. Royal Statist. Soc., 1897, p. 812-54). The biological term "regression"
now had unhelpful connotations and Yule preferred "characteristic lines" to
"regression lines." However, the characteristic terminology did not take.

The terminology and some of the symbols associated with regression
appeared around this time. Pearson (1896) used the phrase "double regression"
but "multiple regression coefficients" appears in the 1903
Biometrika paper "The Law of Ancestral Heredity" by Karl Pearson,
G. U. Yule, Norman Blanchard, and Alice Lee.

The Pearson-Yule notion of regression which appears with correlation
as an aspect of the multivariate normal specification is still current but so
is a second notion developed in the 1920s. R. A. Fisher’s regression specification,
where the dependent variable is normally distributed conditional on the values
of the independent variables, is closer to that underlying Gauss’s first theory
of least squares (see entry on Gaussian). The new regression specification appears
in section 6 (p. 607) of Fisher’s 1922
"The goodness of fit of
regression formulae, and
the distribution of regression coefficients" (J. Royal Statist.
Soc., 85, 597-612), but then to much greater effect in Statistical Methods
for Research Workers. (1925). The opening paragraph of Section 15 on "Regression
coefficients"
(chapter V) has the declaration:

The idea of regression is usually
introduced in connection with the theory of correlation, but it is in reality
a more general, and, in some respects, a simpler idea, and the regression co-efficients
are of interest and scientific importance in many classes of data where the
correlation coefficient, if used at all, is an artificial concept of no real
utility.

This entry was contributed by John Aldrich, drawing on David (1995) and on the description
of Galton’s work in Porter and of the Galton-Pearson-Yule development in Stigler
(1986). See also AUTOREGRESSION, CORRELATION, DEPENDENT VARIABLE, DUMMY
VARIABLE, GAUSS-MARKOV THEOREM, METHOD OF LEAST SQUARES, NORMAL EQUATION,
RESIDUAL, WEIGHT and the discussion of regression notation
on the Symbols in Probability and Statistics page.

REGULAR (as in regular polygon) is found in 1679 in
Mathematicks made easier: or, a mathematical dictionary by
Joseph Moxon, with this definition: "Regular Figures are those where
the Angles and Lines or Superficies are equal." The phrase "regular
curve" occurs in 1665 [OED].

RELAXATION, as a term in numerical analysis for a particular method of successive
approximation, derives from Southwell’s work, beginning
with K. N. E. Bradfield and R. V. Southwell "Relaxation Methods Applied
to Engineering Problems. I. The Deflexion of Beams under Transverse Loading,"
Proceedings of the Royal Society of London A, 161, (1937),
155-181. (From A. S. Householder The Theory of Matrices
in Numerical Analysis (1964, p. 92).)

Gauss proposed a form of relaxation, which led G. E. Forsythe
to write, "Gauss was very fond of relaxation ... [He] remarked that the process
was so easy he could it do while half asleep or while thinking about other things."
(with T. S. Motzkin) "An Extension of Gauss’s Transformation for Improving the
Condition of Systems of Linear Equations," Mathematical Tables and
Other Aids to Computation, 6, (1952), p. 17.

In English, the word was introduced by Robert Recorde, who used
remayner or remainer (Smith vol. 2, page 97).

The term REMAINDER THEOREM appears in 1886 in Algebra by
G. Chrystal [OED].

RENEWAL THEORY, RENEWAL EQUATION, etc. A JSTOR search found the phrase
renewal theory in A. W. Brown "A Note on the Use of a Pearson Type III Function
in Renewal Theory," Annals of Mathematical Statistics, 11, (1940),
448-453. Brown refers back to A. Lotka’s 1939 paper
"A Contribution to the Theory of Self-Renewing Aggregates, With Special Reference
to Industrial Replacement," in the same journal. See the entry SURVIVAL
FUNCTION.

Within a few years this theory of demography for machines
had become part of applied probability, see J. L. Doob’s
"Renewal Theory From the Point of View of the Theory of Probability," Transactions
of the American Mathematical Society, 63, (1948), 422-438.

REPEATING DECIMAL.Circulating decimal is found in December 1768
in the title "On the Theory of Circulating Decimal Fractions" by John Robertson in Phil.
Trans. 58:207.

Recurring decimal fraction is found in December 1768 in
John Robertson, "On the Theory of Circulating Decimal Fractions,"
Phil. Trans. 58:207: "In operations, with such recurring decimal fractions, particularly
in multiplication and division, the work will either be longer than
necessary, or be very inaccurate, if the numbers are not considered
as circulating ones: and to come at the true results of such operations,
several authors have given precise rules; and some of them have shewn
the principles upon which those rules were founded."

Repeating decimal is found in 1773 in the Encyclopaedia
Britannica [OED].

Repeater is found in the 1773 edition of the Encyclopaedia
Britannica: "Pure repeaters take their rise from vulgar fractions
whose denominator is 3, or its multiple 9" [OED].

Infinite decimal is found in 1796
in A Mathematical and Philosophical Dictionary by
Charles Hutton: "Infinite Decimals, such as do not terminate, but go on without end" [OED].

REPETEND appears in 1714 in
A new and compleat treatise of the doctrine of fractions, vulgar and decimal by Samuel Cunn:
"The Figure or Figures continually circulating, may be called a Repetend."

REPLACEMENT SET is dated 1959 in MWCD10.

REPLICATION as a technical term in the design of experiments appears in R.A. Fisher’s
"The Arrangement of Field Experiments,"Journal of the Ministry of Agriculture of Great Britain,33, (1926)
p. 506: "The method [of estimating the standard error] adopted is that of replication."
(OED)

The term REPUNIT was coined by Albert H. Beiler in 1966.

RESIDUAL in a least squares context appears in
1868 in Theoretical Astronomy by James Craig Watson:
"In the case of a limited number of observed values of x, the
residuals given by comparing the arithmetical mean with the several
observations will not ... give the true errors" [OED].

RESIDUE (in complex analysis). Cauchy introduced the concept and the
term résidue in his "Sur un nouveau genre de calcul analogue au Calcul infinitesimal,"
Exercices de Mathématiques 1826, pp. 23-35 in
Oeuvres 2e sér.. Tome VI.
(F. Smithies Cauchy and the Creation of Complex Function Theory.)
The OED’s earliest quotation is from A. R. Forsyth Theory of Functions x. p. 223 (1893)
"The sum of the residues of a doubly-periodic function relative to a fundamental
parallelogram of periods is zero."

Si numerus a numerorum b, c differentiam metitur,
b et c secundum a congrui dicuntur, sin minus, incongrui; ipsum
a modulum appelamus. Uterque numerorum b, c priori in casu
alterius residuum, in posteriori vero nonresiduum vocatur. [If a number a measure the
difference between two numbers b and c, b and c are said to be congruent with respect to
a, if not, incongruent; a is called the modulus, and each of the
numbers b and c the residue of the other in the first case, the non-residue in the latter case.]

Residue has been in
English since the 14th century and as a mathematical word since the
15th but it was originally used to mean remainder. The OED’s
first reference for the word in the modern number theory sense is the report on
number theory by H. J. S. Smith Rep. Brit. Assoc.
Adv. Sci. 1859I. 231.

See MODULUS and REMAINDER.

RESIDUE CLASS appears in 1948 in Number Theory and Its History by
Oystein Ore: "Since these are the numbers that correspond to the same remainder
r when divided by m, we say that they form a residue class
(mod m)" [OED]. A JSTOR search found an article from 1915 by Edward Kircher
"Group Properties of the Residue Classes of Certain Kronecker Modular Systems and
Some Related Generalizations in Number Theory," Transactions of the American Mathematical Society,
16, No. 4. (Oct), p. 413. However the German term would have been coined long before.

The term RESULTANT was introduced by Bézout, according to
Geschichte der Elementar-Mathematik by Karl Fink. The term was
employed by Bézout in Histoire de
l’sAcademie de Paris, 1764, according to Salmon in Modern
Higher Algebra.

Resultant was used by Arthur Cayley in 1856 in Phil.
Trans.: "The function of the coefficients, which, equalled to
zero, expresses the result of the elimination..., is said to be the
Resultant of the system of quantics. The resultant is an invariant
of the system of quantics" [OED].

The term eliminant was suggested by De Morgan, according to
Geschichte der Elementar-Mathematik by Karl Fink.

Eliminant is found in 1881 in Burnside and Panton, Theory
of Equations: "The quantity R is..called their Resultant or
Eliminant" [OED].

REVERSE POLISH NOTATION is found in 1962 in “Translation to and from Polish notation” by C. L. Hamblin in The Computer Journal.
The article states, “Polish notation is so-called because of its extensive use in Polish logical writings since its invention by &Lstrok;ukasiewicz.”

RHODONEA was coined by Guido Grandi (1671-1742) "between 1723
and 1728." He used the Greek word for "rose" (Encyclopaedia
Britannica, article: "Geometry").

RHOMBUS. An obsolete term for rhombus in English was
lozenge, which was used by Robert Recorde in 1551 in
Pathway to Knowledge: "Defin., The thyrd kind is called
losenges or diamondes whose sides bee all equall, but it hath neuer a
square corner" [OED].

Rhombus was first used in English in 1567 by John Maplet in
A greene forest or a naturall historie,...: "Rhombus, a
figure with ye Mathematicians foure square: hauing the sides equall,
the corners crooked" [OED].

The term RHUMB LINE is due to Portuguese navigator and
mathematician Nunes (Nonius) (Smith vol. I).

RICCATI EQUATION is the name given to a differential
equation studied by Vincenzo
Riccati in his “Animadversationes in aequationes differentiales secundi
gradus,” Acta eruditorum Supp. viii. (1724), pp. 66-73. See Klein (ch. 21 “Ordinary Differential
Equations in the Eighteenth Century”) and the entry in the Enyclopedia
of Mathematics.

According to Mathematical Thought From Ancient to Modern Times by Morris Kline, the term Riccati equation was introduced by D’sAlembert.
The reference is Hist. de l’sAcad. de Berlin, 19, 1763, 242 ff. pub. 1770.

An article, “Mr.Murphy’s Solution of Riccati’s Equation,” is found in 1837 in
The Ladies’ Diary. The article is reprinted from the Cambridge Philosophical Society’s
Transactions. [Google print search, James A. Landau]

The matrix Riccati equation came to prominence in the 1960s in connection with optimal
control problems but matrix equations “of the Riccati type” had been studied in
the mathematics literature since the 1930s. See W.
M. Whyburn “Matrix Differential Equations, American Journal of Mathematics, 56,
(1934) 587-592.

RICHARD’S PARADOX was presented as a contradiction in set theory by
Jules Richard
(1862-1956) in a 1905 letter,
"Les principes
des mathématiques et le problème des ensembles"
Revue générale des sciences pures et appliquées, 16, (1905), 541
(translated in Heijenoort (1967)). The contradiction was associated with the
set of numbers that can be defined in a finite number of words. Bertrand Russell
referred to the contradiction as "le paradoxe du Richard" in his
Les Paradoxes de la LogiqueRevue de métaphysique et de morale (1906) p. 638. Peano’s reaction to the contradiction was that
it did not belong to mathematics but to linguistics. It is now classified as
a semantic paradox.

See PARADOX.

RIEMANN HYPOTHESIS. In 1905
Euclid’s Parallel Postulate: Its Nature, Validity, and Place in Geometrical Systems
by John William Withers has: "If we accept Riemann’s hypothesis we cannot
be sure that there will be any such line at all, for
we do not know that space has any infinitely distant parts." [Google print search]

Riemann hypothesis appears in English in its modern sense
in 1924 in the Proceedings of the Cambridge Philosophical Society XXII: "We
assume Riemann’s hypothesis. ... We assume the truth of the Riemann hypothesis" [OED]. The hypothesis is stated by
Bernhard Riemann
(1826-1866) in "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" (1859)
Werke (p. 139)
English translation
(p. 4).

RIEMANNIAN GEOMETRY is dated 1904 in MWCD10.

RIEMANN INTEGRAL appears in 1907 in The theory of functions of a real
variable and the theory of Fourier’s series by Ernest William Hobson:
"The Lebesgue integral of f(x) lies between the upper and lower integrals
of f(x), and is identical with the Riemann integral in case the latter exists."
[University of Michigan Historical Math Collection]

The term RIEMANN SPACE was used by Alfred Clebsch in a lecture delivered on
Aug. 28, 1893: "There is no difficulty in conceiving a four-dimensional Riemann space corresponding
to an equation f(z, y, z) = 0." [Google print search]

RIEMANN SUM is found in 1935 in Interpolation and Approximation
by Rational Functions in the Complex Domain by Joseph Leonard Walsh:
"The sum in the left-hand member of (23) is precisely a Riemann
sum for the integral in the right-hand
member if we have xnk = k/n.
[Google print search]

RIEMANN ZETA FUNCTION. The use of the small letter zeta for
this function was introduced by Bernhard Riemann (1826-1866) as early
as 1857 (Cajori vol. 2, page 278).

RIGHT, OBTUSE and ACUTE ANGLES are the subjects of Definitions
10,
11 and 12
of Book 1 of Euclid’s Elements. The
modern English word right is
derived from Old English while obtuse and acute are derived from the
Latin words for blunt and sharp respectively. See the entry ANGLE.

The OED’s earliest
citation for right angle in English
is from Chaucer’s Treatise on the Astrolabe (c. 1391): “This forseid rihte orisonte..diuideth
the equinoxial in-to riht Angles.” The earliest citation for the other words is
from 1570 and Sir Henry Billingsley’s translation of
Euclid’s Elements: “an obtuse
angle is that which is greater then a right angle" and "An acute
angle is that, which is lesse then a right angle."

RIGHT TRIANGLE.Right cornered triangle is found in
1551 in Pathway to Knowledge by Robert Recorde: "Therfore
turne that into a right cornered triangle, accordyng to the worke in
the laste conclusion" [OED].

Right angled triangle is found in 1594 in Exercises
(1636) by Blundevil: "If they have right sides, such Triangles are
eyther right angled Triangles, or oblique angled Triangles" [OED].

Right triangle is found in 1675 in R. Barclay, Apol.
Quakers: "A Mathematician can infallibly know, by the Rules of
Art, that the three Angles of a right Triangle, are equal to two
right Angles" [OED].

The term rectangular triangle appears in 1678 in Cudworth,
Intell. Syst.: "The Power of the Hypotenuse in a Rectangular
Triangle is Equal to the Powers of both the Sides" [OED].

The term number ring (Zahlring) was coined by David Hilbert (1862-1943) in the
context of algebraic number theory [See
Jahresbericht der Deutschen Mathematiker-VereinigungCapitel IX. Die Zahlringe des Körpers.

Ring is found in English in 1930 in E. T. Bell, “Rings whose
elements are ideals,” Bulletin A. M. S.

[Julio González Cabillón]

RISK and RISK FUNCTION (referring to the
expected value of the loss in statistical decision theory) first appear in Wald’s
"Contributions to the Theory of Statistical Estimation and Testing Hypotheses,"
Annals of Mathematical Statistics,, 10, (1939), 299-326
[John Aldrich, based on David (2001)].

See DECISION THEORY.

The term ROBUSTNESS was used in a 1953 article by George E. P. Box:
"This remarkable property of ’srobustness’s to non-normality which these tests
for comparing means possess, and without which they would be much less appropriate to
the needs of the experimenter, is not necessarily shared by other statistical tests."
Box’s "Normality and Tests on Variances" (Biometrika, 40, 318-335) belonged to a
literature with its origins in the 1920s when W. A. Shewhart and E. S. Pearson first
investigated the behaviour of Student’s t-test and the analysis of variance under non-standard conditions.
(OED2 and David 2001)

According to Cajori (1919, page 224) the term Rolle’s
theorem was first used in 1834 by Moritz Wilhelm Drobisch (1802-1896) and in
1846 by Giusto Bellavitis (1803-1880). Bellavitis used teorema del Rolle in 1846 in the
Memorie dell’s I. R. Istituto Veneto di Scienze, Lettere ed Arte, Vol. III (reprint), p. 46, and again in 1860 in
Vol. 9, section 14, page 187.

Rolle’s theorem is found in English in 1858 in A
treatise on the theory of algebraical equations by John Hymers
[Univesity of Michigan Historical Math Collection].

ROMAN NUMERAL is found in 1735 in Phil. Trans.
xxxix, 139: "The Roman Numeral Ten, which was made in this Form, like an X" [OED].

Also in 1735, The Whole Duty of Man According to the Law of Nature
by Samuel Puffendorf has:
"The Roman Numerals I and II, signifie the First and Second Book."
[Google print search]

ROOT-MEAN-SQUARE is found in Sept. 1895 in Electrician:
"A short time ago Dr. Fleming published a new and ingenious method of plotting wave
forms with polar co-ordinates, and of directly obtaining therefrom the root mean-square value"
[OED].

ROOT of an equation. See RADIX.

ROOT in graph theory. The OED gives A. Cayley in "On the Theory of Analytical Forms
called Trees," Phiosophical Magazine XIII. (1857)
Papers,
II, p. 242 "The inspection of these figures will show at once what is
meant by the term in question, and by the terms root, branches, ... and knots
(which may be either the root itself, or proper knots, or the extremities of
the free branches)."

ROTUNDUM is a Latin word introduced by Peter Ramus (1515-1572)
to refer to the circle or the sphere (DSB).

The term ROULETTE was coined by Pascal (Cajori 1919, page
162). See also cycloid and trochoid.

ROUND (verb; to approximate a number) is found in English in
1840 in Observations on the attempted application of pantheistic
principles to the theory and historic criticism of the gospel by
William Hodge Mill: “300 years: which, after every possible
allowance for rounding the number, will give..a length mostly absolutely
incompatible with the supposition of only two generations.” [OED]

Round up and round down are found in 1956 in G. A.
Montgomerie, Digital Calculating Machines vii. 129: "In a long
calculation, all these increases may accumulate, and it is better to
round some of them up and some of them down" [OED].

RULE OF FALSE POSITION. The Arabs called the rule the hisab
al-Khataayn and so the medieval writers used such names as
elchataym.

RULE OF SUCCESSION. This states that if an event has occurred n times in succession,
then the probability that it will occur again is (n + 1)/(n + 2). This value is the
mean of the posterior density of the probability of a
success given n successes in n Bernoulli trials assuming a uniform
prior for the probability of a success.

John Venn introduced the expression "rule of succession" in his
Logic of Chance (first edition 1865) and
he spent a chapter criticising the rule. It is often called Laplace‘s rule
of succession because it appears in the Introduction Laplace added to the
2nd (1814) edition of his Théorie Analytique des Probabilités.
Laplace also published the introduction separately as the Essai Philosophique
sur les Probabilités.

Laplace’s illustration (p. xvii in the edition on
Gallica) became famous, "if we place the
dawn of history at five thousand years before the present date, we have 1,826,213
days on which the sun has constantly risen in each 24 hour period. We may therefore
lay odds of 1,826,214 to one that it will rise again tomorrow." However, Laplace
continued, "But this number would be incomparably greater for one, who perceiving
in the coherence {or totality} of phenomena the principle regulator of days and
seasons, sees that nothing at the present moment can arrest the course of it."
(from A. I. Dale’s translation of the Essai,Pierre-Simon Laplace:
Philosophical Essay on Probabilities (p. 11, 1995).

(This entry was written by John Aldrich, based on Dale op. cit. and two articles by S. L. Zabell (1988):
"Buffon, Price and Laplace: Scientific Attribution in the 18th century,"
Archive for History of Exact Sciences, 39, 173-181 and (1989)
"The Rule of Succession," Erkenntnis, 31, 283-321.)

The term RULE OF THREE was used by Brahmagupta (c. 628) and by
Bhaskara (c. 1150) (Smith vol. 2, page 483).

From Smith (vol. 2, pp. 484-486):

Robert Recorde (c. 1542) calls the Rule of Three "the
rule of Proportions, whiche for his excellency is called the Golden
rule," although his later editors called it by the more common name.
Its relation to algebra was first strongly emphasized by Stifel
(1553-1554). When the rule appeared in the West, it bore the common
Oriental name, although the Hindu names for the special terms were
discarded. So highly prized was it among merchants, however, that it
was often called the Golden Rule, a name apparently in special favor
with the better mathematical writers. Hodder, the popular English
arithmetician of the 17th century, justifies this by saying: "The
Rule of Three is commonly called, The Golden rule; and indeed
it might be so termed; for as Gold transcends all other mettals, so
doth this Rule all others in Arithmetick." The term continued in use
in England until the end of the 18th century at least, perhaps being
abandoned because of its use in the Church.

Rule of three appears in English in 1562 in Well Sprynge Science by H. Baker:
“The rule of three is the chiefest, the moste profitable, and the moste excellente rule of all the rules of Arithmetike.” [OED]

Numerous 18th- and 19th-century wills and other documents which can
be found on the Internet require that certain persons should learn
arithmetic "to the rule of three."

Abraham Lincoln (1809-1865) used rule of three in an
autobiography he wrote on December 20, 1859:

There were some schools, so called; but no qualification
was ever required of a teacher beyond "readin, writin, and cipherin"
to the Rule of Three. If a straggler supposed to understand latin
happened to sojourn in the neighborhood, he was looked upon as a
wizzard. There was absolutely nothing to excite ambition for
education. Of course when I came of age I did not know much. Still
somehow, I could read, write, and cipher to the Rule of Three; but
that was all. I have not been to school since. The little advance I
now have upon this store of education, I have picked up from time to
time under the pressure of necessity.

Charles Darwin (1809-1882) wrote:

I have no faith in anything short of actual measurement
and the Rule of Three.

[The Darwin quotation was provided by John Aldrich, who points out
the interesting fact that Lincoln and Darwin were born on the same
day, February 12, 1809.]

The term RUNGE-KUTTA METHOD (named for
Carle Runge and
Wilhelm Kutta)
apparently was used by Runge himself in 1924,
according to Chabert (p. 441), who writes:

RUSSELL’S PARADOX. Bertrand Russell
(1872-1970) found the contradiction (as he called
it) in 1901. Chapter X of his Principles of Mathematics (1903) is called
"The Contradiction." Ernst Zermelo (1871-1953) discovered the paradox in 1899
but he did not publish it. See Grattan-Guinness (2000, p. 216)